\(\int \frac {(a+b x)^5}{x^{10}} \, dx\) [93]

Optimal result

Integrand size = 11, antiderivative size = 67 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {a^5}{9 x^9}-\frac {5 a^4 b}{8 x^8}-\frac {10 a^3 b^2}{7 x^7}-\frac {5 a^2 b^3}{3 x^6}-\frac {a b^4}{x^5}-\frac {b^5}{4 x^4} \]

[Out]

-1/9*a^5/x^9-5/8*a^4*b/x^8-10/7*a^3*b^2/x^7-5/3*a^2*b^3/x^6-a*b^4/x^5-1/4*b^5/x^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {a^5}{9 x^9}-\frac {5 a^4 b}{8 x^8}-\frac {10 a^3 b^2}{7 x^7}-\frac {5 a^2 b^3}{3 x^6}-\frac {a b^4}{x^5}-\frac {b^5}{4 x^4} \]

[In]

Int[(a + b*x)^5/x^10,x]

[Out]

-1/9*a^5/x^9 - (5*a^4*b)/(8*x^8) - (10*a^3*b^2)/(7*x^7) - (5*a^2*b^3)/(3*x^6) - (a*b^4)/x^5 - b^5/(4*x^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^5}{x^{10}}+\frac {5 a^4 b}{x^9}+\frac {10 a^3 b^2}{x^8}+\frac {10 a^2 b^3}{x^7}+\frac {5 a b^4}{x^6}+\frac {b^5}{x^5}\right ) \, dx \\ & = -\frac {a^5}{9 x^9}-\frac {5 a^4 b}{8 x^8}-\frac {10 a^3 b^2}{7 x^7}-\frac {5 a^2 b^3}{3 x^6}-\frac {a b^4}{x^5}-\frac {b^5}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {a^5}{9 x^9}-\frac {5 a^4 b}{8 x^8}-\frac {10 a^3 b^2}{7 x^7}-\frac {5 a^2 b^3}{3 x^6}-\frac {a b^4}{x^5}-\frac {b^5}{4 x^4} \]

[In]

Integrate[(a + b*x)^5/x^10,x]

[Out]

-1/9*a^5/x^9 - (5*a^4*b)/(8*x^8) - (10*a^3*b^2)/(7*x^7) - (5*a^2*b^3)/(3*x^6) - (a*b^4)/x^5 - b^5/(4*x^4)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85

method	result	size
norman	\(\frac {-\frac {1}{4} b^{5} x^{5}-a \,b^{4} x^{4}-\frac {5}{3} a^{2} b^{3} x^{3}-\frac {10}{7} a^{3} b^{2} x^{2}-\frac {5}{8} a^{4} b x -\frac {1}{9} a^{5}}{x^{9}}\)	\(57\)
risch	\(\frac {-\frac {1}{4} b^{5} x^{5}-a \,b^{4} x^{4}-\frac {5}{3} a^{2} b^{3} x^{3}-\frac {10}{7} a^{3} b^{2} x^{2}-\frac {5}{8} a^{4} b x -\frac {1}{9} a^{5}}{x^{9}}\)	\(57\)
gosper	\(-\frac {126 b^{5} x^{5}+504 a \,b^{4} x^{4}+840 a^{2} b^{3} x^{3}+720 a^{3} b^{2} x^{2}+315 a^{4} b x +56 a^{5}}{504 x^{9}}\)	\(58\)
default	\(-\frac {a^{5}}{9 x^{9}}-\frac {5 a^{4} b}{8 x^{8}}-\frac {10 a^{3} b^{2}}{7 x^{7}}-\frac {5 a^{2} b^{3}}{3 x^{6}}-\frac {a \,b^{4}}{x^{5}}-\frac {b^{5}}{4 x^{4}}\)	\(58\)
parallelrisch	\(\frac {-126 b^{5} x^{5}-504 a \,b^{4} x^{4}-840 a^{2} b^{3} x^{3}-720 a^{3} b^{2} x^{2}-315 a^{4} b x -56 a^{5}}{504 x^{9}}\)	\(58\)

[In]

int((b*x+a)^5/x^10,x,method=_RETURNVERBOSE)

[Out]

1/x^9*(-1/4*b^5*x^5-a*b^4*x^4-5/3*a^2*b^3*x^3-10/7*a^3*b^2*x^2-5/8*a^4*b*x-1/9*a^5)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)^5/x^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=\frac {- 56 a^{5} - 315 a^{4} b x - 720 a^{3} b^{2} x^{2} - 840 a^{2} b^{3} x^{3} - 504 a b^{4} x^{4} - 126 b^{5} x^{5}}{504 x^{9}} \]

[In]

integrate((b*x+a)**5/x**10,x)

[Out]

(-56*a**5 - 315*a**4*b*x - 720*a**3*b**2*x**2 - 840*a**2*b**3*x**3 - 504*a*b**4*x**4 - 126*b**5*x**5)/(504*x**
9)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)^5/x^10,x, algorithm="maxima")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)^5/x^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^5}{x^{10}} \, dx=-\frac {\frac {a^5}{9}+\frac {5\,a^4\,b\,x}{8}+\frac {10\,a^3\,b^2\,x^2}{7}+\frac {5\,a^2\,b^3\,x^3}{3}+a\,b^4\,x^4+\frac {b^5\,x^5}{4}}{x^9} \]

[In]

int((a + b*x)^5/x^10,x)

[Out]

-(a^5/9 + (b^5*x^5)/4 + a*b^4*x^4 + (10*a^3*b^2*x^2)/7 + (5*a^2*b^3*x^3)/3 + (5*a^4*b*x)/8)/x^9